Invariant submanifolds for homogeneous flows on quotients of semisimple Lie groups of noncompact type.

Abstract

Let ${\cal M}$ be a compact locally symmetric space of noncompact type. Let ${\cal N}$ be an immersed compact submanifold of $S{\cal M}$ that is invariant under the geodesic flow. It is shown that in the case that the rank of ${\cal M}$ is one, ${\cal N}$ is of the form $S{\cal M}\sp\prime$, where ${\cal M}\sp\prime$ is a compact immersed totally geodesic submanifold of ${\cal M}$. In the case that ${\cal M}$ has rank two or greater, similar but weaker conclusions are drawn. It is also shown that if there is a totally geodesic isometric immersion of a simply connected symmetric space of noncompact type into ${\cal M}$, the closure of the image in ${\cal M}$ is a compact totally geodesic submanifold of ${\cal M}$. Techniques of Lie groups, Pesin Theory and Ratner's Theorem are used in the proofs.